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No. 2009–66
LAST TIME BUY AND CONTROL POLICIES WITH PHASE-OUT RETURNS: A CASE STUDY IN PLANT CONTROL SYSTEMS
By Harold Krikke, Erwin van der Laan
August 2009
ISSN 0924-7815
1
Last Time Buy and control policies with phase-out returns:
a case study in plant control systems
Harold Krikkea and Erwin van der Laan
a. Corresponding author, CentER, Tilburg University,
PO box 90153, 5000 LE, Tilburg, The Netherlands, [email protected]
b. RSM Erasmus University,
Department of Decision and Information Sciences,
PO box 1738, 3000 DR, Rotterdam, The Netherlands, [email protected]
2
Abstract: This research involves the combination of spare parts management and reverse logistics. At the
end of the product life cycle, products in the field (so called installed base) can usually be serviced by either
new parts, obtained from a Last Time Buy, or by repaired failed parts. This paper, however, introduces a
third source: the phase-out returns obtained from customers that replace systems. These returned parts
may serve other customers that do not replace the systems yet. Phase-out return flows represent higher
volumes and higher repair yields than failed parts and are cheaper to get than new ones. This new
phenomenon has been ignored in the literature thus far, but due to increased product replacements rates its
relevance will grow. We present a generic model, applied in a case study with real-life data from ConRepair,
a third-party service provider in plant control systems (mainframes).
Volumes of demand for spares, defects returns and phase-out returns are interrelated, because the
same installed base is involved. In contrast with the existing literature, this paper explicitly models the
operational control of both failed- and phase-out returns, which proves far from trivial given the non-
stationary nature of the problem. We have to consider subintervals within the total planning interval to
optimize both Last Time Buy and control policies well. Given the novelty of the problem, we limit ourselves to
a single customer, single-item approach. Our heuristic solution methods prove efficient and close to optimal
when validated.
The resulting control policies in the case-study are also counter-intuitive. Contrary to (management)
expectations, exogenous variables prove to be more important to the repair firm (which we show by
sensitivity analysis) and optimizing the endogenous control policy benefits the customers. Last Time Buy
volume does not make the decisive difference; far more important is the disposal versus repair policy. PUSH
control policy is outperformed by PULL, which exploits demand information and waits longer to decide
between repair and disposal. The paper concludes by mapping a number of extensions for future research,
as it represents a larger class of problems.
Key words: spare parts, reverse logistics, phase-out, PUSH-PULL repair, non stationary, Last
Time Buy, business case
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1. Introduction This paper discusses how service repair firms apply reverse logistics in supplying spare parts for
servicing aging mainframe plant control systems. Plant control systems are used for automated
process management of large industrial installations. In the final stages of the mainframe’s
product life cycle, spare parts become scarcer and expensive to produce. Repair is a good
option, but failed parts cannot always be repaired. This paper therefore introduces a new source
that can be tapped to achieve this: phase-out returns. These are returns retrieved from systems
that are being abandoned (phased out) by one customer, but may still fit another customer in a
second life. In particular for aging products such as mainframes, phase-out returns are used to
guarantee availability of spare parts for products still in operation. Due to today’s replacement
rates we often face multiple so-called phase-out occurrences and supply of phase-outs can be
abundant. Often these returns are in still good (‘repairable’) condition.
In the service process, three major actors play a role. Original Equipment Manufacturers
(OEMs) fulfil targeted service levels to their customers at a certain price. Customers include oil
refineries and platforms, offshore windmills, medical facilities and nuclear power plants, all high-
value capital goods. Failures of plant control systems are rare, but incur tremendous downtime
cost when they happen. So a high availability of spare parts is needed.
At some point, OEMs outsource the service to the already introduced third party repair
firm, the focal company of this study. This service provider then supports customers on behalf of
the OEM for the remaining product life time (usually several years). General terms and conditions
(i.e., service levels and prices) on both servicing customers who keep the old (mainframe)
systems and scheduling of phase-outs of others are formalized in an Extended Service Program
(ESP), which is a deal between the OEM and the repair firm. The repair firm takes over full
operational and financial responsibility for service repair of the mainframes. Based on the ESP
the repair firm makes Service Level Agreements (SLAs) with individual customers, concerning
similar but possibly differentiated terms and conditions.
The timing of outsourcing to the repair firm usually coincides with the conclusion of the
OEMs production of new spares. It then offers the (one-time) opportunity to acquire a final batch
of new service parts to the repair firm: the so-called Last Time Buy (LTB). Alternatively, the repair
firm may want to use future phase-out returns as a cheaper but more uncertain source for spares.
The repair firm balances between own (profit driven) interests and
customer service. In this balancing act, often repair firms apply a combined
control policy for repair and disposal decisions and LTB. A major issue is to
decide to repair or scrap repairable returns immediately upon arrival (PUSH) or
4
on demand (PULL). In our opinion, decision support models can make the tradeoffs more
visible. Figure 1 summarizes the problem at hand schematically.
Figure 1 about here
Triggered by a real-life business case (ConRepair), this paper investigates and evaluates the
modelling and optimization of a combined LTB & control policy, with the additional option of
phase-out returns. By documenting a real life case study we identify this new phenomenon, which
has thus far been ignored in the literature, and whose relevance is increasing with shortening
product life cycles. Phase-outs complicate the analysis considerably, since they reduce the
installed base and with it, change the demand- and return rates over time. At the start of the
planning period the demand for repairs will be relatively high and phase-out volumes low, but this
changes over time. Clearly, the time-based matching of supply and demand is complicated as the
problem is non-stationary. In optimizing LTB & control policies one cannot simply rely on net
demand throughout the planning interval as most models do. Sub-intervals are necessary. We
adopt a simplified (single item, single customer) heuristic approach as a first step. We use a cost
driven model where lacking service is translated into backorder cost and penalties. The aim is to
maximize overall profit for the (focal) repair firm, using generic LTB and control policies.
The case results in turn have consequences for the generic policies; as they are
counterintuitive. In fact (endogenous) decisions of the focal firm turn out to have little impact on
the overall result. ConRepair depends on exogenous variables to maximize its profits, while the
endogenous policy variables serve the customers’ interest. Optimizing LTB volume does not
make the difference; rather, it is the repair/disposal control policy that is of major consequence.
We show that PUSH is outperformed by PULL, which exploits demand information as it waits
longer to decide.
Section 2 proceeds by developing the business case of ConRepair, and introduce the
problem. Based on this, Section 3 reviews the relevant literature in order to obtain building blocks
and motivate our modelling approach. Section 4 gives a (mathematical) problem description,
including assumptions and delineations. We then develop an appropriate model to analyze the
issues at hand. Section 5 validates the heuristics used and presents the results of an elaborate
numerical study using ConRepair data. Finally, Section 6 contains a discussion and the
conclusions. The results and limitations of our research are used to indicate possible follow-up
research.
2. Business practice
5
The OEM in this case study is Honeywell Industrial Automation and Control (IAC), a global player
in industrial automation. It produces and maintains Distributed Plant Control Systems, i.e.
networks of intelligent (automated) mainframe stations that control an industrial (chemical) plant.
Because of the high capital value of these plants involved and their dependency on control
systems, the latter require prompt service in case of failure.
Nowadays, desktop-based plant control systems serve increasingly as an alternative to
the existing mainframes in the field. Customers, of course, replace their mainframe systems at
their own discretion, and some do that more quickly than others. Hence, the phase-out return of
one customer (who introduces desktops) may well serve another customer as a spare part for
mainframes that stay in operation. The installed base, i.e., all products in operation in the field,
now provides two sources of returns (defects and phase-outs) and also serves as a reuse market
(demand for spares). Due to the phase-outs, its size reduces over time.
Returned parts are repaired and made available again as spare parts, complementary to newly
acquired spares. Phase-out returns represent larger volumes than failed parts, as both return
rates and yield (the percentage of the return flow that is repairable) are higher.
Honeywell IPC has outsourced the service of controllers of -amongst others- HPEP,
Excel EMC, DGP and Mikronik- to ConRepair. ConRepair was established by a management
buy-out (MBO) from the former Honeywell Amsterdam Repair Center in 1997. The company
oversees the financing, planning and execution of the entire support process, dealing directly with
the final customer, based on the ESP.
ESPs concern the service process for “mission-critical” parts only. The repair firm commits it self
to collecting and testing all defect and phase-out returns. We deal with expensive parts but failure
incurs even higher collateral (downtime) cost. The repair company jockeys the pooling advantage
using commonality between customers to both improve availability and reduce cost.
The time-period covered by an ESP is negotiated, but is usually 3 to 5 years. The contract
period represents our planning interval. When the ESP agreement between OEM and repair firm
ends all service is terminated. Based on the ESP, one develops individual SLA contracts between
the final customers and the repair firm. An SLA defines minimal service requirements fulfilled by
the repair firm and prices to be paid by customers. Availability from stock (either new- or repaired
parts) and the total reliability (including delivery backorders within at most one week) is
measured. New parts cost the full catalogue list price, while repaired parts generally cost less
(about 80% of list price), but usually also have lower cost. Companies like ConRepair generate
income by charging the prices in case of actual failure; thus, if no failures occur ConRepair has
no income, but still all of the cost. The repair firm pays the OEM for the LTB, which represents a
major investment at the start of the ESP planning interval. Although largely exogenous, phase
outs scheduling is essential. To be of use in covering demand for spares, phase-out collection
6
must take place (just) before failures. Because phase-outs decrease the installed base’ size (of
mainframes) smaller, demand for spares decreases during the planning interval and phase-out
volumes increase. This essential input of the control policy. As mentioned, the basic options for
control policies in practice are PUSH and PULL. In the literature, e.g. van der Laan et. al (1998), it
is described in the following manner:
1. PULL policy: all repair activities are done reactively as soon as the serviceable inventory
position drops below the base stock level. This policy is expected to keep serviceable
inventories small, but there is more dependency on the inbound repairables inventory.
2. PUSH policy: all repair activities are done proactively, upon arrival of a part return. This
policy is expected to have larger outbound inventories (whereas there are no inbound
inventories).
PUSH is more intuitive, and has more flexibility to schedule repairs conveniently as they are not
directly needed for service. However, it also bears the risk of repairing parts that prove unneeded
later on. PULL creates a time advantage by waiting for actual demand and delays financial
investments (read repair cost) until actually needed. But it introduces additional lead-time for
repair when serviceable inventory is too low. PULL operates in practice as a base stock level with
one-for-one repair; repair starts whenever a demand arises. Please note that PUSH is not a
special case of PULL, even if the base stock level is set arbitrarily high, since PUSH is triggered
by product returns and PULL is triggered by product demands. Contrary to PUSH, PULL builds
up an inbound inventory of repairables. Take-back of phase-out parts increases the possibility of
oversupply and hence the need for scrapping repairables prematurely, i.e. scrap upon arrival.
Similar decisions such as LTB-volume may have different optimal values for PUSH and PULL.
Next to the management of repair operations, a disposal policy controls the inflow of part
returns. An (also optimized) dispose-down level stipulates the point at which repairable inbound
returns are to be scrapped instead of repaired— namely, when the total number of parts in the
system is sufficient to cover expected future net demand (demand minus future returns).
Scrapping excessive returns averts the costs of keeping stock, but also removes the possibility to
repair at a later stage. Non-repairables are scrapped immediately at all times.
To determine the Last Time Buy (LTB), the customers forecast the expected number of
failed parts based on historic failure rates. The repair firm has to forecast the repair yields.
Although so called planning interval is fixed in the ESP, uncertainty is caused by the demand for
parts in the future, which depends on the failure rates and (on the supply side) the number as well
as timing and repair yield of future phase-out and defect returns needed to cover future demand.
3. Literature
7
The literature on spare parts management is abundant and may be divided into two parts:
scheduled maintenance and unplanned maintenance (Kennedy et al. 2002). For the latter, safety
stocks need to be in place in order to be able to deal with future maintenance activities. Typical
decisions that need to be made are whether to replace or repair on site, and where to
strategically place inventories of spare parts. Since the Last Time Buy decision, the repairability
of failed parts and the non-stationarity of demand due to phase-outs are relevant for our case,
this review focuses on (combinations of) those aspects.
Papers in the field do not typically deal with drops in demand as the installed base size is
stable. In that sense, our research is related to the literature on product obsolescence, where it is
commonly assumed that, whereas demand is stable during the lifetime of a product, the
remaining lifetime itself is a random variable (David et al. 1997). In our case, however, the
remaining lifetime of a product is fixed and known due to ESP, although demand may drop due to
planned phase-outs. Cattani and Souza (2003) acknowledge the importance of the Last Time Buy
(in their case, ‘end-of-life build’) decision, and focus on its optimal timing for a fixed and known
remaining number of periods of demand. Building on earlier results (Cattani 1997), Cattani and
Souza (2003) quantify the benefits of delaying the end-of-life build. In our research, the timing of
the Last Time Buy decision is fixed because it concurs with the moment of outsourcing of OEM to
repair firm. Also the period which the repair firm must provide service to the customer is a given.
But in our case demand for spare parts over the planning horizon is non-stationary, as the
installed base changes over time, due to phase-outs. To achieve optimality, the control policy
must deal with balancing supply and demand, which enhances the importance of the scrap
versus repair decision.
Few studies combine Last Time Buy and repair/disposal policies, let alone phase-outs.
Below we describe those who have come closest to our problem. Teunter and Klein Haneveld
(1998) describe a situation in which an OEM stops supplying spare parts for a single machine.
The operator of the machine is offered an opportunity to place final orders for a number of critical
parts to keep the machine operational up to a fixed horizon. Based on this horizon, the failure
rates of the components, the prices of spare components, holding costs and machine downtime
costs, the size of the (near-)optimal final orders are determined. Unused parts are scrapped only
at the end of the service period, which means that such a policy may lead to high stocking costs.
In another paper, Teunter and Fortuin (1999) consider a more complicated situation in which
failed spare parts can be repaired and reused, and unused parts can be removed from stock
before the end of the horizon using a dispose-down-to level. Since the cost of repair is assumed
to be negligible, all returned items are repaired and re-stocked. Some of the theoretical results
were applied at Philips, as described in Teunter and Fortuin (1998). Our case is similar to the one
in Teunter and Fortuin (1999), although we do not assume that repair costs are negligible nor that
8
all parts can be repaired. Repair cost reduce the viability of immediate repair (read PUSH policy),
particularly when subject to a quality based yield factor as is the case in our study.
The above-mentioned papers constitute a category that resorts to newsvendor-type
approaches to determine (near-) optimal final orders. In contrast, Spengler and Schröter (2003)
take a system dynamics approach in order to take into account product life-cycle aspects and
dependencies between sales and returns. Rather than focusing on finding optimal final orders,
they investigate whether and how parts reuse can contribute to spare parts management after the
final order has been placed. They also explore certain system dynamics.
The next category of relevance is the case of seasonal products with limited
replenishment options. Although they lack a repair process, returns are explicitly modelled. Many
products, such as apparel, are demanded during a short period of time only, while replenishment
opportunities are very limited or non-existent. Mostard and Teunter (2006) analyze the case of a
Dutch mail-order company. A single order is placed before the selling season starts. Purchased
products may be returned by the customer for a full refund within a certain time interval. Returned
products are resalable, provided they are returned before the end of the season and are
undamaged (this equals a yield factor in the sense that the fraction of returns that can be reused
is fixed and known). Products remaining at the end of the season are disposed of. All demands
not met directly are lost. A simple closed-form newsvendor equation determines the optimal order
quantity given the demand distribution, the probability that a sold product is returned, and all of
the relevant revenues and costs. Although Teunter and Fortuin (1999) and Mostard and Teunter
(2006) explicitly model product returns, and Spengler and Schroter (2003) control repair and
disposal implicitly through recovery and recycling rates, the repair process is not very explicitly
modelled.
A final relevant body of literature concerns models with combined new manufacturing and
repair operations where control policies deal with an inbound repairables inventory and an
outbound serviceable inventory (for an overview, see van der Laan et al. 2004). The latter
consists of both new and repaired items, which are assumed to be identical. Convenient policies
to control the repair process are so-called PUSH policies (part returns are repaired as soon as a
certain quantity is reached) and PULL policies (a certain quantity from part-return inventory is
repaired as soon as the inventory position drops below a certain value). Depending on the cost
structure and lead times, either PUSH or PULL will perform better (van der Laan 1998, van der
Laan 1999). No last time buy applies to these models.
All of the above papers take a cost-driven approach, meaning that falling short on service
levels can be given a monetary value by means of backorder- and penalty costs. Service-level
approaches (optimizing service levels regardless of the costs) are described in general terms
(without returns) in Fortuin (1980, 1981), Sherbrook (2004) and Muckstad (2005). Service
models are more intuitive to customers, but more difficult to use because a constant monitoring
9
and calculation is needed on the achieved performance levels. At each point in time we must
check whether we should start repair operations. This aspect demonstrates the major drawback
of a service-level approach, which is simultaneously a ‘look back’ and a ‘look ahead’ policy. Its
mere look-ahead nature makes the cost approach more robust than service-level models, which
are inclined to overcompensate past drops in performance. Also it allows for unambiguous
interpretation of results as it is unilaterally monetary.
We do not know of any papers that combine the last time buy decision with explicit PUSH
and PULL repair decisions with limited repair yield (of two streams). We are the first to introduce
the phase-out phenomenon into the problem. The solutions as suggested in the past literature do
not suffice to deal with this situation, because in order to exploit the phase-outs we must model
volume, timing and yield of both return types explicitly because their behaviour is totally different.
Also we must connect phase-out occurrences to installed base size and hence demand for
spares. This paper develops control policies and heuristic control rules to facilitate the integral
management of acquiring, repairing and disposing spare parts, while monitoring the performance
throughout the remaining service time after the last time buy.
4. A modelling approach
4.1 Problem description, assumptions and simplifications
The installed base, of size IB(t) at time t, is serviced through a pool (serviceable inventory ) of
identical spare parts that are either new or repaired. Demand fulfilled from serviceable stock,
consists of new (unit cost cm) and repaired (unit cost cr) parts. Scrapping of unneeded or non-
repairable parts is assumed to incur zero cost. New parts are used up first because of their higher
capital value; the use of repaired parts takes place at an increasing pace over time. There is no
quality difference between new and repaired parts. Although engineers have other duties, such
as calibration of new systems, repair gets priority once repairables are available and repair is
needed, so we linearize unit repair costs and to set repair capacity to be infinite.
The customers that operate the installed base are identical in the sense that they have
negotiated the same conditions and prices for new (pm) and repaired (pr) parts as well as similar
service levels. The presence of an ESP as a foundation for individual SLAs makes it feasible to
model the total installed base as a single meta-customer.
In line with general ESPs, failures are fixed at most one week after the initial call—
considering that, on average, redundancy in the mainframes keeps systems up and running for
about that period of time. Repair lead-time is assumed to be one week and fixed. Available parts
are delivered from the serviceable inventory immediately; otherwise the customer has to wait for
the part during the (fixed) repair lead-time,
10
To monetarize service levels, low responsiveness is penalized. I*f no serviceable stock is
available, then the demand is backordered, and a penalty fee cb per part per week is paid to the
customer to compensate for the delay. Backorders that are still outstanding at time t=H cannot be
fulfilled anymore and are penalized with cf per part (cf >>cb). Hence, penalty costs cb provides an
incentive to offer a certain availability of serviceable parts throughout the planning horizon
through an optimized repair policy. Penalty cost cf provides an incentive to guarantee a certain
availability at the end of the planning horizon through an optimal choice of the LTB.
Part failures (a Poisson process with failure rate in the installed base both generate
demand for spare parts and cause the return of the failed parts. The service provider has to place
a final order for a specific part at time t=0, but it has contractual agreements to provide service for
that part until time t=H. Then at time t=0 it needs to optimally choose Last Time Buy quantity, Q,
and its future repair policy taking into account stochastic future demand and part failures and the
deterministically scheduled phase-outs in order to optimize its profits during the time interval
[0,H]. Since all demand is triggered by failed parts in the installed base, demand is itself a
Poisson process with rate )(tIB .
Standard procedure would be to determine the LTB based on projected net demand over
the entire planning interval [0,H], But with phase-outs this may not always lead to sufficient
performance, as the following simplified example shows. Consider a deterministic world with an
initial installed base of 100 parts, continuous demand for spare parts (4 parts per time unit) over
the planning horizon (100 time units), but parts cannot be repaired. However, there is one phase-
out occurrence of 50 parts after 50 time units (so halfway) and they are all repairable. After the
phase-out occurrence the installed base halves and therefore the demand drops to 2 parts per
time unit. The net demand for spare parts over the planning horizon equals demand prior to the
phase-out plus demand after the phase-out minus the phase-out returns=50x4+50x2-50=275.
The LTB could therefore be chosen to be 250 and no stock-outs would occur. Note that the LTB
is sufficient to handle the demand (200 parts) prior to the phase-out occurrence. Now suppose
that instead of halfway, now after 75 time units (so later) a phase-out of size 75 (instead of 50)
occurs. In that case we have the same net demand 75x4+25x1-75=250, but now an LTB of 250
parts is no longer sufficient to cover the demand prior to the phase-out occurrence (4x75=300
parts).
In other words: the non-stationarity of the problem and the fact that demand until the first
phase-out occurrence may exceed net demand complicates the optimization of both LTB and
operational control. Therefore we have to split the total planning interval into subintervals and
formulate minimal availability for each one of them. Otherwise we bear the risk of (huge)
temporary drops in service level. One way that is effective and tractable within our modelling
framework is to specify a maximum acceptable stock-out probability, 1-p just before a phase-out
occurrence.
11
At ConRepair we observed that phase-outs are scheduled by customers. Therefore, in
our model we assume that the timing of phase-out returns is deterministic and known at the
beginning of the planning period. As these phase-outs reduce the size of the installed base,
future demand for spare parts will also be reduced. Both phase-out returns and returns of failed
parts can be repaired, but they may have different expected repair yields (yP and yF, respectively).
To control the repairable inventory N(t) and serviceable inventories of acquisitioned parts M(T)
and repaired parts R(T), we employ either a PUSH policy (repair upon arrival) or a PULL policy
(at any time t repair parts, as long as the inventory position— that is, serviceable inventory
M(t)+R(t) minus backorders plus repair work in-progress— drops below base-stock level S(t)).
The fixed repair lead time equals L. As at some points in time the number of available parts may
be deemed sufficient to service the projected future net demand, it may be cost-effective to
dispose of some of the incoming repairable parts. To accommodate this, we introduce the
decision variable U(t): any incoming part will be disposed of if the echelon inventory position
(Inventory of repairables plus inventory position) is at or above U(t).
In order to maximize expected profits over the interval [0,H] we need to develop rules that
optimize decision variables Q, S(t) and U(t), which is the objective of the remainder of this
section. The complete list of notations and base-case data is given in Table 1. Note that data are
confidential and therefore normalized. A schematic representation of the model is given in Figure
2. The complexity makes a heuristic approach needed. As even the heuristic problem poses quite
some challenges we limit ourselves to a single-item, which however does not harm the generic
nature of the model when assuming that demand for individual spares is independent. We also
assume perfect information on all parameters, even in sensitivity analysis, meaning that all
decisions in LTB and control policy can be adapted when input values change. Exogenous
variables such as failure rates, repair yields and variable (unit) operating costs and revenues are
constant in time. To explain the model in a lucid manner, we start with a stationary situation
without phase-out returns.
Figure 2 about here
4.2 No phase-outs
Even without phase-outs, the analysis is complicated by the fact that there are three different
stocking points: for newly acquired parts, repaired parts and repairable parts with carrying charge
(hm , hr and hn, respectively). Schematically, if we ignore the repair lead-time L (which is
reasonable, as L is very small compared to planning horizon H), then the inventory processes
under PUSH and PULL control follow the behaviour depicted in Figure 3.
12
Figure 3 about here
Since there are no phase-outs, the installed base is fixed ( )0()( IBtIB for all t) and weekly
demand is therefore stationary, with expectation )0()( IBDE . Under PUSH control, the
inventory of new parts decreases from the initial level Q with the average demand rate until all
parts are used (say, at time X). Until that moment, repaired parts have accumulated to the
expected value RQy (Q demands cause Q returns that are repaired upon arrival with expected
yield RQy ). As soon as the new parts have run out, demand is serviced through repaired parts.
The serviceable stock decreases with the net demand (demand minus repairable returns) rate.
Under PULL control, returned parts accumulate in the repairable inventory N(t) until at some time
t the inventory position IP(t) falls below level S(t).Then a sufficient number of repairable products,
if available, is ordered for repair in order to restore the inventory position to level S(t). If the stock
of repairables is not sufficient to meet this level, then all of the available parts are repaired. To
limit the number of decision variables, we assume zero fixed costs for repair, so that no batching
takes place for economies of scale.
As long as repairables are available, the PULL policy operates as an (S(t)-1,S(t)) policy. If
demand during lead-time follows a normal distribution with cdf G(.), which is reasonable since the
underlying processes are Poisson processes, then in the long run the near-optimal base stock
level S*(t) satisfies (Silver et al. 1998, p. 255)
),(),(1)( * LttDVarkLttDEtS ,
where
),(11
LttDVarGk
(1)
and nrb
b
hhc
c
(Silver et al. 1998, p. 266).
The purpose of the Last Time Buy quantity Q is to stock sufficient acquired parts at the beginning
of the planning period such that stock-out costs at the end of the planning horizon are well
balanced against the operational costs during the planning horizon. If there were no phase-outs
to consider, we could have formulated the problem as a standard newsvendor problem. Given a
certain realization, z, of net demand over the planning horizon (i.e. zHND ),0( ), the profit
function is given as (see Silver et al. 1998, p. 404)
13
QzQzBpQQv
QzzQgpzQvzQP
,)(
,)(),(
where v denotes the cost per unit acquired, p denotes the revenue per unit sold, g denotes the
salvage value for any unit not sold by the end of the planning horizon, and B denotes the penalty
cost for demand not satisfied at the end of the planning horizon. In order to fit our problem in the
standard newsvendor formulation, appropriate values of v, p, g, and B need to be developed.
Note that v should somehow reflect the inventory carrying costs. Theory tells us that the expected
profit function reads as
)),0(Pr(1)()(),0()()( QHNDBgpQgvHNDEgpQPE ,
and is optimized for the Q that satisfies
Bgp
BvpQHND
),0(Pr . (2)
Under PULL control, the repairable inventory reaches its maximum as soon as the inventory of
newly acquired items, Q, is depleted at time t=X. Just before that time, at about )(/ DESX ,
where E(D) equals expected weekly demand, the repair facility starts processing repairable parts
up to level S(t) (see Figure 3b). Assuming that inventories linearly increase/decrease with the
return rate and (net) demand rate, the total inventory carrying costs are approximately given by
HShyhXHSDE
SHhyh
H
XQ
hxHSDESHhySQhXQ
nFrnFm
rnFm
2
1
)(22
1
)()(/2
1)(
2
1
2
1
2
2
(3)
Timepoint )(/ DEQX obviously depends on Q, which makes optimization problematic, since
total carrying costs are quadratic in Q. However, the fraction of demand that is fulfilled by newly
acquired parts, HX / , is approximately equal to 1 minus the fraction of demand that is
potentially serviced through part repairs, or FyHX 1/ . Using this approximation, the
inventory carrying cost contributed through Q during time interval [0,H] is then approximated by
2/))1(( HhyhyQ nFmF and should be included in the cost per acquired unit v.
14
Turning back to our newsvendor formulation, appropriate values for v, p, g, and B are as follows:
2/))1(( Hhyhycv nFmFm - unit acquisition cost + carrying cost per unit acquired
mpp - unit sales price of new parts
0g - surplus is scrapped against zero profit/cost.
fcB - shortage is penalized against cf per unit short
Equation (2) then becomes
fm
nFmFm
cp
HhyhycQHND
2/))1((
1),0(Pr . (4)
Under PUSH control, all repairables are ‘pushed’ to the serviceable inventory upon arrival, so that
the total inventory carrying costs simply read
HhyhH
XQHhQyhXQ rFmrFm
2
1
2
1
2
1.
Through FyHX 1/ this is approximately equal to 2/1 HhyhyQ rFmF , so that
under PUSH control 2/))1(( Hhyhycv rFmFm . This expression is similar to that under
PULL control with hn simply replaced by hr. Hence, as the values of p, g, and B remain the same,
under PUSH control equation (4) holds with hn replaced by hr.
In order to prevent the build-up of excessive stocks of repairables, we adopt the following
disposal policy: do not accept incoming returns as long as the echelon inventory position
IP(t)+N(t) equals or exceeds level U(t). In order to derive near-optimal values of U(t), we analyze
the impact of accepting an incoming return at time t through a marginal analysis. Suppose we
accept an incoming return, and IP(t)+N(t) is raised to U(t). The marginal investment vt of
accepting the return should cover the marginal benefit of accepting it (sales value p +avoided
penalty cf times the probability of selling it), so that
))(),(Pr(1)( tUHtNDcpv ft ,
or
f
t
cp
vtUHtND
1))(),(Pr( (5)
15
for appropriate values of vt and p. Under PUSH control, the consequence of accepting an
additional return at time t is repair cost cr and total carrying cost )( tHhr , so
)( tHhcv rrt . If the part is sold at the end of the planning horizon, it generates value
rpp . Under PULL control you invest in repairing the part only when you expect to sell it, so
)( tHhv nt and rr cpp . This rule adjusts the desirable inventory of serviceable and
repairable parts as more information (past demands, returns and repaired parts) becomes
available. Note that one should dispose of all incoming returns whenever HtLH , since
they cannot be repaired before the end of the planning horizon. The values of vt and p are
summarized in Table 2.
Table 2 about here
4.3 Full model with phase-outs
Phase-outs complicate matters in two ways: 1) demand is no longer stationary over the planning
interval [0,H], as the demand rate decreases after each phase-out occurrence (demand is
stationary though in between phase-out occurrences), and 2) it is no longer valid to determine Q
solely on the basis of the net demand during the interval [0,H]. To see this, suppose that phase-
outs are planned at times nii ,..,1, . If a phase-out is scheduled relatively early, then Q could
be calculated solely on the basis of the net demand during the interval [0,H] (see Figure 4a),
since during the planning interval [0,H] sufficient parts are available to ensure reliable operations.
But if a phase-out is scheduled near t=H, this same Q may not be sufficient to satisfy demand up
to the phase-out occurrence (see 5b).
Figure 4 about here
One way to get around this issue would be to specify the maximum allowable risk, i.e. stock-out
probability (1-p), that management is willing to accept just before a phase-out occurrence at time
i . Based on this maximum allowable risk we calculate iQ for each interval [0, i ]. Taking the
maximum over all i ensures that at each it the stock-out probability is at most (1-p). Section
5.1 elaborates on the proper setting of p.
Although the inventory process is more complicated when compared to the case without
phase-outs, we choose to use the same approach as in the previous section— that is, proceeding
16
from a particular timepoint X onwards, allow inventories to reduce gradually with the average net
demand rate.
If we let Hn 1 and 00 , then over the complete planning horizon H the Last Time
Buy quantity Qn+1 should satisfy
fm
nn cp
vQND
1),0(Pr 11 , (6)
where under PULL control 2/)( Hhyhcv nFsm , and is an approximation of the
fraction of demand that is fulfilled by newly acquired parts:
1
01
1
01
)()(
)()0()()(
1n
jjjj
P
n
jjjjF
IB
HIBIByIBy
.
Note that for n=0 we have )()0( HIBIB , so reduces to )1( Fy and equation (6) reduces
to (4). For ni 1 , we choose to satisfy the following conditions in order to maintain a minimum
performance just before phase-out moments i :
Pii QND ),0(Pr , ni 1 .
Under PUSH control, the same expressions hold— with hn simply replaced by hr. The near-
optimal value of the Last Time Buy quantity is then calculated as
ii QQ max .
In the same vein we determine the dispose-down-to level at time t for planning horizon H:
f
tn cp
vtUHtND
1))(),(Pr( 1 (7)
and
Pii tUtND )(),(Pr , ni 1 ,
where vt, and p are given, as in Table 2. The near-optimal value of U(t), finally, is calculated as
itLi tUtUi
)(max)( | .
17
The above-mentioned heuristic decision rules applied together should minimize total net profit
NP, defined as total sales minus total acquisition and repair costs, minus total inventory carrying
costs, minus total penalty costs. The next section first assesses the quality of our approximate
rules by comparing them with optimal solutions obtained through enumeration. Subsequently, we
analyze our policies for the base case of Table 1 and various other scenarios.
5. Numerical study
The base-case scenario for our numerical study is based on actual data from ConRepair,
collected over the period 2004-2007 for a particular part (see Table 1). For reasons of
confidentiality, the cost data are normalized and the part name is not specified. We assume that
parts fail according to a negative exponential distribution with failure rate . Hence, between two
sequential phase-outs i and 1i the demand (and return) process is a Poisson process with
mean )( iIB . For the heuristics it is convenient to assume that net demand during some time
interval is normally distributed. This assumption is justified, since between two sequential phase-
outs i and 1i also net demand is a Poisson process, with mean )()1( iF IBy .
We apply the PUSH and PULL heuristics as developed in the previous section, analyzing
several variations of the base-case scenario through simulation. Each simulation run consists of a
period of three years (150 weeks). For each scenario, the simulation was run 3000 times. The
observed maximum relative error in the net profit over all reported simulations was below 1%.
Section 5.2 reports the base case and describes the impact of exogenous changes in repair yield
and the size and timing of phase-outs. Results are represented in terms of (net) profit,
contribution of the LTB, disposal rates and availability of spares. First, however, we validate the
developed heuristics in Section 5.1.
5.1 Model validation
As a result of model complexity, we had to resort to a number of approximations and heuristic
procedures. Although optimal solutions (obtained, for instance, by extensive (enumerative)
simulations) may be preferable, this can only be done for Last Time Buy quantity Q and never for
all control policy decisions. Moreover, the derived formulas provide us an easy handle with which
we can interpret and understand the numerical results, since they are more lucid and closer to the
actual decision-making practice. Finally, there is the practical advantage of a considerable
amount of time saved, compared to enumerative simulation. A drawback is that the approximation
of the expected inventory may perform poorly, especially when phase-out volumes are large. The
18
heuristics should therefore be of sufficient quality. This we will check by comparing heuristic and
optimal values of Q.
Figure 5 shows the actual behaviour of the inventory processes for the base case (see
Table 1) under our heuristic policies and near-optimal decisions. It can be seen that these are
very similar to the stylized pictures of Figure 3 that were used as a basis for the inventory cost
approximations.
Figure 5 about here
In order to evaluate the accuracy of the Last Time Buy heuristics, we disable the disposal option
and initially set 0p . Table 3 shows that the heuristics for the Last Time Buy quantity are very
accurate for the complete range of possible repair yields. The PULL heuristic slightly
underestimates the Last Time Buy, but this hardly affects performance; all scenarios of Table 3
show heuristic Last Time Buy quantities that differ less than 2% from the optimal ones, while net
profits differ less than 1% from optimal. Considering that this is comparable to simulation
inaccuracies, the differences in net profit are probably not statistically significant. Increasing the
size of the Last Time Buy decreases the length of the period that parts are repaired and thus the
probability of stock-outs under PULL. The PULL heuristic does not take this into account, thereby
underestimating the Last Time Buy. The performance of the PUSH and PULL policies depends
on the availability of repairable products, and is therefore sensitive to stock-outs of repairable
parts that may occur just before a phase-out. The performance will therefore depend on the
appropriate setting of p (see Table 4). An alternative to the service-level approach in setting
p is to follow exactly the same cost-based control rules that are used for Hn 1 for all i ,
and then to take the maximum values of Qi and Ui(t) After some experimentation, we found that
the qualitative results of such an approach are identical to the service-level approach. For ease of
presentation, we therefore adhere to the service-level approach.
Table 3 also shows that the differences between PUSH and PULL regarding Last Time
Buys are rather small, which suggests that inventory carrying costs are only a secondary
determinant of the Last Time Buy quantity. Indeed, expression (4) claims that the main trade-off is
between the unit acquisition cost, on the one hand, and the unit sales price and unit penalty cost
at the end of the planning horizon, on the other— unless the planning horizon is very large.
Tables 3-4 about here
Evaluating the accuracy of the disposal policy is rather difficult, as we do not know the structure
of the optimal policy. Its performance, however, can be compared against policies without
19
disposal. Table 5 shows the value of disposal for varied phase-out timing. In general, disposal
appears to be beneficial, and its value increases as the phase-outs move further towards the end
of the planning horizon. PUSH benefits much more from a disposal policy, as it is punished more
severely for excessive stocks than PULL is. Expression (6) predicts that the disposal rate
increases with higher inventory carrying charges, higher unit repair costs and lower unit sales
costs of repaired parts. Based on (7), it is easily shown that for practically all relevant scenarios
( rr pc ,and frn cHhHh ) the disposal level is generally lower under PUSH than under
PULL, which means that PUSH, on average, disposes of more product returns than PULL does.
Table 5 about here
5.2 Base-case results
Table 6 reports on the performance of the PUSH and PULL heuristics for the base-case scenario.
In terms of net profit per week, the PULL policy is superior to the PUSH policy. This is due mainly
to the difference in total inventory costs and penalty costs. PULL’s emphasis on less inexpensive
repairable inventory, rather than on repaired inventory, provides an important cost advantage.
One might expect a firm to consider a PUSH policy in order to guarantee a higher service level,
but results show that this is not the case. The explanation is that to reduce the investment in
expensive repaired inventory, the Last Time Buy quantity Q is suppressed and/or more returns
are disposed of. The probability of stock-outs then increases, resulting in service levels that are
lower than under PULL. Actually, none of the many scenarios that we analyzed yielded a better
performance for PUSH, as compared to PULL.
Table 6 about here
5.2.1 Impact of the unit repair cost
An important determinant of the performance of PUSH and PULL is the unit repair cost.
Intuitively, if repair is cheap, PUSH may be a reasonable option. If repair is expensive, it may be
better to use PULL. Figure 6 shows that while PUSH always performs worse than PULL does, its
relative performance becomes better for smaller cr. Here we modelled the carrying charge for
repaired parts as rnr cwhh , with w the opportunity cost of capital, in order to reflect the
financial impact of stocking repaired parts. So, hr increases linearly in cr. It is easily shown that
when cr=0 (and thus hr= hn), PULL is equivalent to PUSH. As cr grows it becomes increasingly
attractive to temporarily stock returns before repair until they are really needed.
Figure 6 about here
20
5.2.2 Impact of the repair yield of failed parts
There is, of course, a time dependency between demands and returns (in the sense that a
demand generates a return at the same moment), but the lead-time implies that one cannot
benefit directly from that relation. In order to satisfy a demand directly from stock, a demand has
to be fulfilled through a new part or a repaired part that had been returned in the past. It appears
from Figure 7 that if the repair yield of failed parts rises, then the net profits increase— albeit less
than linearly. Although the Last Time Buy volume naturally goes down with increased
repairability, one has to rely on a return flow that is stochastic in timing and yield, which means
that the Last Time Buy volume decreases less than linearly. It is important to note that
serviceable inventory stock-outs can occur only once all new parts are used up. With decreasing
Last Time Buy volume, the time tr that all demand is satisfied directly from stock through new
parts goes down too, so that the risk of stock-outs increases. This is also explained through
relation (8), which shows that service level goes down as tr goes down. As the repair yield
approaches 1.0, virtually all demand is satisfied from repairs, so there is no time for accumulating
safety stocks of parts. Service level therefore decreases sharply. The total fraction of demand
delivered ( ), however, remains high overall, so that customers can be confident— even if they
are more dependent on returns. Note that the difference between PUSH and PULL diminishes as
the repair yield approaches 1, since there is hardly any opportunity to stock parts.
Figure 7 about here
Analysis with the repair yield of phase-out shows similar results. Intuitively, one would expect that
phase-outs, being a cheap source of supply, would enhance both service levels and net profits
with increasing yield. Unfortunately, this is only partially true: too high yields lead to carelessness
of the control policy regarding safety stock levels.
In case both the repair yields would be zero, the Last Time Buy would have to fully cover all
demand for spares, and ConRepair’s operations for this part would no longer be profitable (note
that demand will still go down). Next to the repair yield of the phase-out returns, their volume and
timing proves even more important as the next section shows.
5.2.3 Impact of phase-out volume
Next, we vary the volume of phase-out returns (Figure 8 and 10). As most of the demands are
fulfilled through repairables, this is quite an important scenario. Observe, first of all, that net profit
decreases with phase-out volume (Figure 8a). Reductions in the installed base decrease total
demand for spare parts over the planning horizon, so that net profits decrease as well. The net
profit per unit demand initially increases up to an annual installed-base reduction of 30% (Figure
8b)), since the returns coming from the phase-outs replace expensive new parts (hence, Q
21
decreases). At some point, however, the Last Time Buy cannot decrease further, since it has to
protect against demand leading up to the first phase-out. An increasing volume of phase-out
returns needs to be disposed of, and the relative contribution in fulfilling demand by returns goes
down, forcing marginal profits down. This explains the convex curve of relative Last Time Buy
contribution in Figure 9b. The diminishing role of returns also explains why the difference
between PUSH and PULL decreases with higher annual reductions in the installed base. The
superior performance of PULL is reconfirmed, however, as it waits for the right moment of repair
and fewer returns are scrapped prematurely.
The managerial conclusion would be that phase-outs might be a source for returns, but
that the effect of reduced turnover overrules all effects. In other words: no phase-outs may be
best for ConRepair, since the installed base then remains of maximal size. But phase-outs are a
given, so preventing or reducing them may not be an option. Next, we investigate the impact of
the timing of phase-outs.
Figures 9, 10 about here
5.2.4 Impact of phase-out timing (and frequency)
Spreading the phase-outs (that is, increasing the frequency of phase-out moments into smaller
batches, whilst keeping the total phase-out volume stable) hardly affects performance, so we will
not report any further details here. A possible explanation is that inventory costs are relatively
low, and if parts are returned in time and are not scrapped prematurely, there will be sufficient
repairables available to meet demand. This result also strengthens the idea that phase-out
returns make the difference mainly at the end of the period H, and failed parts fulfil much of the
early demand, despite lower yield, because they are more equally spread. We therefore
investigate the effect of changing the timing of an individual phase-out event.
In the scenario of Figure 10 there is one phase-out occurrence, and its timing is varied from week
25 to week 125. Phase-outs reduce the installed base, and thereby demand and net profits.
Delaying phase-outs maintains the installed-base size and thereby total profit. It is even more
instructive to look at net profits per unit demand to investigate the impact of phase-out timing.
Delaying phase-outs at first increases net profits per unit demand, because the arrival of the
phase-out returns allows the firm to become more and more aligned with the demand for repaired
parts. Profits improve, due to lower holding costs. At some point, though, the phase-outs come in
too late, more and more returns need to be disposed of and marginal profits start to decrease.
Total profit keeps increasing, but the curve flattens. The managerial implication is that it pays off
to delay phase-outs somewhat, although marginal profits decrease.
22
In this scenario the gap between PUSH and PULL is the largest when the role of returns
is smallest— that is, when phase-outs occur either early (little demand, hence fewer failures) or
late (excessive phase-out returns are disposed of) in the planning period.
Figure 10 about here
5.2.5 Wrap up
PULL is better at matching supply and demand because it waits longer to decide and uses actual
demand information. PUSH often scraps too early or repairs too quickly. PULL, at the end of the
day, leads to lower operational costs and due to better service has fewer problems with penalties
and backorders. Last Time Buy quantity hardly differs between PUSH and PULL. The role of the
scrap versus repair decision outweighs the Last Time Buy volume. So control policies should be
PULL-driven repair combined with a dispose-down-to level geared optimally toward controlling
the inbound returns.
A main paradox in our study is that customer vigilance leads to a service-level focus on
their behalf, whilst repair firms want to maximize profits. Results show however that the
profitability of phase-out use for ConRepair is low, because customers get a discount. Profits for
ConRepair could be boosted by increased prices for repair. At the moment, all cost benefits are
transferred to the customer: whereas the quality of repaired parts is equal to new parts, the price
is only 80%. An alternative is to reduce repair cost leading to the same effect.
At first sight, therefore, there doesn’t seem to be much point in take-back and repair of
phase-outs. Lower phase-out volumes are actually beneficial to the firm because the installed
base remains larger and the demand for spares stays high. Although an exogenous variable, this
may be communicated to the customer (and OEM). Managerially this means that merely the
scheduling of phase-outs can be adapted. It is essential that phase-outs are available as late as
possible but in time to be reused.
There is another exogenous factor that has a big impact on profitability: the repair yield of
both failed parts and phase-out returns. Once collection timing is optimized, collection quality
must be ensured (by e.g. decent packaging), in order to avoid transport damage. Although
endogenous variables can optimized by PULL for the repair firm, the exogenous variables have a
stronger impact on the profitability.
6. Conclusions and outlook
This paper studies the way in which (mainframe) plant control systems can be serviced using
phase-out returns. The case study at hand represents a larger class of problems. Not only are
23
ESPs & SLAs of increasing importance in service logistics because downtime of installed bases
is increasingly costly. Shortening product life cycles and replacement rates will boost phase-out
returns. The global service and support market is growing. In 2010 it will represent a potential
turnover of 90 billion US$ per year (Blumberg, 1999, 2005). Improving performance in this field,
as presented in this case, represents millions of dollars.
Practically all OEMs in capital goods at some point stop servicing and new spares
production. Customers may abandon the mainframes phase out their systems, causing phase-out
returns. At the same moment OEMs outsource to a 3rd party repair firm for customer who keep
the old products, and offer Extended Service Programs (ESPs) for the remainder of the life cycle.
At the start of these programs, OEMs generally offer the opportunity for the Last Time Buy (LTB)
to the repair firm. An LTB of new parts can cover demand for spares for the given ESP period.
Alternatively repairing both failed parts and particularly phase-out returns is possible as well.
Repair is cheaper, but it introduces uncertainty. Service-level Agreements (SLAs) define the
relationship between individual customers and repair firm.
Many parameters are written down in SLAs, but some trade-offs aspects remain difficult.
This is clarified by introducing decision modelling. Contrary to existing literature, we model both
types of returns as one of them (phase-outs) directly impacts the size of the installed base,
leading to a non-stationary system. A dispose-down level is needed because oversupply of
returns is likely to occur. We introduce subintervals to deal with net demand before phase-out
occurrences. We also explicitly model the repair process with yield factors and optimize inbound
and outbound inventories with PUSH –PULL policies. The model translates lacking performance
into backorder and penalty costs. We prefer this to the service level approach because financial
consequences of malperformance can be modelled unilateral with other cost. We develop an
efficient heuristics model based on cost-level optimization.
Policies are optimized in a single customer, single-item situation with subintervals. The
approximate decision rules prove to be both efficient and close to optimal. The resulting policies
are not trivial, but in general PULL policies perform better, if well optimized, because they wait
longer to decide. Postponing repair and scrapping of returns pays of, due to exploiting actual
information on the dynamic installed base, related failure rates and the inbound inventory of
(phase-out) repairables.
Customers strive for maximal certainty, whilst repair firms (such as ConRepair) aim for
profit optimization. But the SLA and corresponding policies achieve the opposite effect. From the
perspective of the repair firm, endogenous control policy variables primarily optimize customer
service and exogenous variables mainly determine the company’s profit. It is remarkable, that
cost benefits are passed on fully to the customer. This calls for a reconsideration of the ESPs
terms and conditions. Formalizing decision making in a model makes trade offs more visible, and
we recommend that all parameters, such as penalty structures, should be part of an SLA.
24
Another suggestion for further research is to extend the model into multi-item situations.
Also, one could investigate, for example, options for selective, condition-based take-back,
applying local testing at the customer site. Moreover, all sensitivity analysis in this paper is carried
out presuming a priori information about parameter changes. Under these circumstances, policies
can be adapted in time. But suppose that information becomes available after the Last Time Buy
has been acquired. Does the presence of phase-outs strengthen robustness of the system, or
does it instead constitute another source of uncertainty itself? Another dimension is differentiation
of service levels into e.g. platinum, gold and economy contracts. It would be interesting to
investigate how control policies are influenced by such differentiation. Finally, optimization
procedures could be extended with exact solutions, service-level models or hybrid PUSH-PULL.
In this, a ‘fair’ calculation of penalty costs, based on collateral cost in the supply chain or
insurance fees, is a fruitful area.
In general, the interaction and synergy of service - and reverse logistics deserves more
exploration. This paper is a first step.
Acknowledgements
We would like to express our gratitude to dr M.C. van der Heijden of Twente University of
Technology, Netherlands, for his critical comments and good advices.
This research is partially sponsored by Transumo ECO project, proj. no. GL05022b.
25
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27
Table 1: Model parameters and base-case data (retrieved 2004-2007)
Model inputs
H Planning horizon (weeks) 150 weeks
IB(t) Installed base at time t (parts) 0≤t< 50 : 500 parts
50≤t<100 : 400 parts ( 501 )
100≤t<150 : 300 parts ( 1002 )
Failure rate (parts per week; total failures follow a piecewise
homogenous Poisson process with rate )(tIB
0.02 parts per week
L Repair lead time (weeks; deterministic) 1 week
hm Carrying cost for newly acquired parts 0.042 per part/per week
hr Carrying cost for repaired parts 0.026 per part/per week
hn Carrying cost for repairable parts 0.010 per part/per week
pm Price of newly acquired part 10 per part (normalized)
pr Price of repaired part 8 per part (normalized)
cm Unit cost of newly acquired part 8 per part (normalized)
cr Unit cost of repaired part 4 per part (normalized)
yF Probability that repair is successful for failed parts 0.7
yP Probability that repair is successful for phase-out parts 0.9
cb Backorder penalty per part 5 euro per part per week
cf Fail-to-deliver penalty per part 100 euro per part
1-p Acceptable probability of being out-of-stock just before
phase-out occurrence, specified by management
0.01
Control decision variables
Q Last Time Buy Quantity (PUSH and PULL)
S(t) Repair-up-to level at time t (PULL)
U(t) Dispose-down-to level at time t (PUSH and PULL)
System process variables
D(a,b) Demand during time interval (a,b] (Poisson process)
R(a,b) Repairable returns during time interval (a,b] (Poisson process)
ND(a,b) Net demand (=D(a,b)-R(a,b)) during time interval (a,b] (Poisson process)
M(t) Inventory of newly acquired parts at time t
R(t) Inventory of repaired parts at time t
N(t) Inventory of repairable parts at time t
IP(t) Inventory position at time t (= M(t) + R(t) + repair in progress - backorders)
28
Table 2: Parameters that determine the disposal quantity under PUSH and PULL
PULL PUSH
tv
p
)( tHhn
rr cp
)( tHhc rr
rp
Table 3: Accuracy of the PULL and PUSH heuristics for varied repair yield and 0p .
Disposal option is disabled; all other parameters are according to the base case.
PULL PUSH
YF
Q
(heuristic)
Q
(optimal)
Net profit
(heuristic)
Net profit
(optimal)
Q
(heuristic)
Q
(optimal)
Net profit
(heuristic)
Net profit
(optimal)
0.3 701 707 13.10 13.28 700 700 9.38 9.38
0.5 457 466 20.52 20.69 456 456 17.55 17.55
0.7 212 222 27.06 27.27 210 211 24.83 24.84
0.9 49 50 23.54 23.54 49 49 23.18 23.18
Table 4: Performance of the PULL and PUSH heuristics as a function of service-level
requirement p ; all other parameters are according to the base case.
PULL PUSH
p Q Net profit Q Net profit
0.900 212 27.06 211 24.84
0.990 225 27.25 225 25.20
0.995 229 27.21 229 25.25
0.999 238 27.06 238 25.19
29
Table 5: The value of disposal for varied timing of the first (and only) phase-out
occurrence 1 .
PULL PUSH
1
Net profit
(disposal)
Net profit
(no disposal)
Net profit
(disposal)
Net profit
(no disposal)
25 22.70 22.70 19.75 19.90
50 25.11 25.12 22.50 22.58
75 26.53 26.40 24.38 22.09
100 27.35 27.12 25.18 21.06
125 27.82 27.70 25.24 19.97
Table 6: Performance of PUSH and PULL in terms of costs and benefits per week.
PULL PUSH
Sales new 15.00 15.00
Sales repaired 51.94 51.87
Production/purchase costs 12.00 12.00
Repair costs 26.16 26.01
New part. Inv. Costs 0.76 0.76
Repaired part Inv. Costs 0.19 1.72
Repairables Inv. Costs 0.35 -
Backorder costs 0.04 0.17
Failed to deliver costs 0.17 1.00
Subtotal 66.94 39.68 66.87 41.67
Net profit 27.25 25.20
Q 225 225
Percentage of returns disposed 3.6% 4.3%
Fraction of parts delivered
directly from stock
99.65% 99.45%
Fraction of parts delivered 99.98% 99.88%
30
31
Serviceable inventory
Last Time
Buy usual
replenishment
Phase-out and defect returns
Repair
Scrap
customer site
depot
Figure 1: The service process and reverse logistics with Last Time Buy opportunity
32
Figure 2: Schematic representation of the model
Last Time Buy
Repairprocess
Inventory of acquired parts, M(t)
cm
Inventory of repairable parts, R(t)
Inventory of repairable parts, N(t)
L, cr
Failures with rate IB(t) and yield yF
pm
pr
hm
hr
hn
St
DisposalUt
cb , cf
Scheduled phase-outs with yield yP
Q
Installed base, IB(t)
33
Figure 3: Inventory processes as a function of time under PUSH (a) and PULL (b) control without phase-outs
(a) (b)
Q
X H
QF
S
Q
X H
X – S/E(D)
(Q-S)F
Repaired parts
0 0
M(t)
R(t) N(t)
M(t)
R(t)
34
Figure 4: The effect of phase-outs under PUSH control: an early phase-out (a) and a late phase-out (b).
(a) (b)
Q
X H
phaseout
M(t)
R(t)
1
Q
X H
phaseout
M(t)
R(t)
1
35
Figure 5: Numerical example for base-case scenario under PUSH (a) and PULL (b) control
(a) (b)
36
Figure 6: Performance of PULL and PUSH as a function of the unit repair cost cr
37
Figure 7: Impact of repair yield
(a) (b)
38
Figure 8: Financial impact of phase-out volume
(a) (b)
39
Figure 9: Impact of phase-out volume with respect to Last Time Buy quantity and service levels
(a) (b)
40
Figure 10: Impact of phase-out timing
(a) (b)
